Unitary fusion categories are not very well understood. We have examples coming from finite groups, and from quantum groups, but not much more. There's been recent progress on the `quadratic' fusion categories, generating a number of new examples. There are also the fusion categories coming from the extended Haagerup subfactor, which remains quite mysterious. It's unclear whether we have so few examples because there aren't many examples, or because it's hard to find them. I'll give an update on the classification of small index subfactors, and what it says about this question.
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